ONT Re: Identity & Teridentity
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
I&T. Note 15
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Re: Quine's supposed "bisection of the triad".
I get the feeling that maybe Lyris is playing the role
of Descartes' Demon's Evil Twin here, so I will recopy
my earlier posting with the quotation from Quine.
This remark, its equivalents in other places, and
the 1953 "Reduction to a Dyadic Predicate" paper,
are the ones that most folks cite on this issue.
I will copy out the 1953 paper later on, when
I figure out how to asciify it, but the first
thing that I mark in his ostentatious 2-adic
predicate F is this big "V" sign -- Pynchon
fans take note! -- for disjunction, or as
Quine would call it, "alternation", right
smack dab in the middle of it. But later.
| Relations in the sense here considered are known, more particularly,
| as 'dyadic' relations; they relate elements in pairs. The relation of
| giving (y gives z to w) or betweenness (y is between z and w), on the other
| hand, is triadic; and the relation of paying (x pays y to z for w) is tetradic.
| But the theory of dyadic relations provides a convenient basis for the treatment
| also of such polyadic cases. A triadic relation among elements y, z, and w might
| be conceived as a dyadic relation borne by y to z;w [the ordered pair (z, w)].
|
| Quine, 'Math Logic', p. 201
|
| W.V. Quine,
|'Mathematical Logic, Revised Edition,
| Harvard University Press, Cambridge, MA, 1981.
With Quine's text in view, I can now add to my list of reasons
why the basic facts of 3-adic irreducibility and 3-identity have
continued to remain such a bother outside of mathematics and most
areas of computer science, where they have been considered trivial
observations since the time of Euler in the first case and Peirce
in the second, at the very least.
JA: Most of the controversy in other circles
appears to turn on (1) not understanding
the statement of the question, as it is
generally understood, and as Peirce most
definitely understood it -- this appears
to be the problem with Quine's fallacy,
since what he does prove is irrelevant
and trivial with respect to the matter
in question, (2) failing to define the
terms that one is using in a way that
makes the problem well-posed.
What Quine knew and when he knew it is not the business of my inquiry here.
I think that it is fair to refer to Quine's statement as "Quine's Fallacy"
because of the uses he and others have put it to, and because, if he knew
better, he simply did not take up the reponsibility or making that clear.
Reason (3), that Quine so amply exemplifies at this point, is this:
Not understanding what a relation is. I know that probably sounds
shocking, so let me explain. We find a category of thinkers who
are perfectly capable of saying what a relation is, speaking in
extension, as is our concern here, they will quite facilely say:
| A k-adic relation is a set of k-tuples, a subset L c X^k, for
| an inclusive enough domain X and its k^th cartesian power X^k.
So far so good.
But when they come to speak on matters like the "composition",
the "decomposition", the "production", or the "reduction" of
a given relation in relation to a given set of relations, they
constantly fail to draw the correct conclusion about what that
means. To facilitate the remainder of this discussion, let us
introduce the generic terms "(de-)generation" to range over all
of the above (de-)constructions in the obvious way. Then, the
immediate consequence that they fail to appreciate is just this:
| A relation is a set of tuples.
| ----------------------------------------------------------------
| A generation of a relation is a generation of a set of tuples.
I will take it up from there next time.
Once again, taken in extension, which is sufficient, and at any rate
what Peirce was talking about -- since a consistent comprehension is
one that is capable of having an extension, there is really no wffle
room to be had by taking that old intensional dodge -- the definition
is that k-adic relations are SETS of ordered k-ples, not just ordered
k-ples simpliciter, and certainly not "ordered sets". Therefore, what
it means to de-compose a 3-adic relation into a composition of 2-adic
relations is to de-compose a set into a composition of two other sets.
And what that means is given by a definition that all mathematically
literate people know. And it happens to be an important fact, about
as important as knowing that 2, 3, 5, 7, 11, 13, and an infinitely
large number of others to be named later, are prime numbers in
ordinary arithmetic, maybe even moreso.
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o